From explicit estimates for the primes to explicit estimates for the M\"obius function II

Abstract

We improve on all the results of [13] by incorporating the finite range computations performed since then by several authors. Thus we have \begin{align*} \Bigg|\sum_{n\le X}\mu(n)\Bigg| &\le \frac{0.006688\,X}{\log X},&&\text{for } X\ge 1\,798\,118, \\\Bigg|\sum_{n\le X}\frac{\mu(n)}{n}\Bigg| & \le \frac{0.010032}{\log X},&& \text{for } X\ge 617\,990. \end{align*} We also improve on the method described in [13] by a simple remark.